Deformation and breakup of bridges of Newtonian and non-Newtonian fluids held captive between two disks that are separated from one another at a constant speed are studied computationally. When the liquid bridge is at the incipience of breakup. a thin liquid thread connects two large volumes of fluid that are pendant from and sessile on the top and bottom disks. High viscosity and elasticity are known from experiments to lead to formation of long threads: these are precursors of satellite droplets which are usually unwanted in applications such as ink-jet printing. To investigate the role of shear-thinning in suppressing long threads and to separate the effect of elasticity from shear-thinning, the rheology of non-Newtonian fluids is described here by a Carreau model which simply accounts for sheer-thinning behavior. When the dynamics is axially symmetric with respect to the common axis of the bridge and the disks, the physics is described by a spatially two-dimensional (2-D) theory. In addition to this fully 2-D theory. a one-dimensional( 1-D) theory based on the slender-jet approximation is also developed here. Both the 2-D and 1-D problems are solved by a method of lines employing the finite element method for spatial discretization and an adaptive finite difference technique for time integration. The computational results show that the limiting bridge length L-d at breakup increases with increasing stretching speed U for both Newtonian and shear-thinning fluids. However, in the case of high-viscosity bridges, as compared to a Newtonian fluid with viscosity equal to the zero sheer-rate viscosity of a shear-thinning fluid, the rate at which L-d of a shear-thinning fluid varies with U becomes less pronounced as U increases. Furthermore, in the case of low-viscosity bridges, the axial location along the thread at which the bridge breaks switches from the vicinity of the bottom of the bridge to its top and then back to its bottom again as U is increased. This switch in the breakup location has important implications in determining the fate of satellite droplets if any are formed. It is also shown that both the shape of the bridge and that of the liquid thread are profoundly affected by shear-thinning behavior, 1-D models have of course been previously used but often without direct comparison to experimental measurements or predictions made with exact 2-D models. It is shown here for the first time that 1-D models are remarkably accurate at low stretching speeds but fail at high stretching speeds. Furthermore, it is demonstrated that as the bridges thin, the dynamics in the vicinity of the location where the bridge radius is smallest follow scaling laws recently developed by others who have analyzed the local behavior of the governing equations close to pinch-off.